Geometric & Functional Analysis GAFA

, Volume 4, Issue 4, pp 373–398

Embedding Riemannian manifolds by their heat kernel

  • P. Bérard
  • G. Besson
  • S. Gallot
Article

Abstract

By embedding a class of closed Riemannian manifolds (satisfying some curvature assumptions and with diameter bounded from above) into the same Hilbert space, we interpret certain estimates on the heat kernel as giving a precompactness theorem on the class considered.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [B1]P. Bérard, Spectral geometry: direct and inverse problems, Springer Lecture Notes in Math. 1207, 1986.Google Scholar
  2. [B2]P. Bérard, From vanishing theorems to estimating theorems: the Bochner technique revisited, Bull. Amer. Math. Soc. 19 (1988), 371–406.Google Scholar
  3. [BBesG]P. Bérard, G. Besson, S. Gallot, Une inégalité isopérimétrique qui généralise celle de Paul Levy-Gromov, Invent. Math. 80 (1985), 295–308.Google Scholar
  4. [BG]P. Bérard, S. Gallot, Inégalités isopérimétriques pour l'équation de la chaleur et applications à l'estimation de quelques invariants, Séminaire Goulaouic-Meyer-Schwartz, exposé no. 15 (1983–84).Google Scholar
  5. [BeGaM]M. Berger, P. Gauduchon, E. Mazet, Le spectre d'une variété riemannienne, Springer Lecture Notes in Math. 194, 1971.Google Scholar
  6. [Bes1]G. Besson, A Kato type inequality for Riemannian submersions with totally geodesic fibers, Annals of Global Analysis and Geometry 4:3 (1986), 273–289.Google Scholar
  7. [Bes2]G. Besson, On symmetrization, in “Nonlinear Problems in Geometry” (D. De-Turck, ed.), Contemporary Mathematics 51 (1986), 9–21.Google Scholar
  8. [C]I. Chavel, Eigenvalues in Riemannian Geometry, Academic Press, New-York 1984.Google Scholar
  9. [CdV]Y. Colin de Verdière, Sur la multiplicité de la première valeur propre non nulle du laplacien, Comment. Math. Helv. 61 (1986), 254–270.Google Scholar
  10. [Co]G. Courtois, Spectrum of manifolds with holes, Preprint (June 1992)Google Scholar
  11. [F]H. Federer, Geometric measure theory, Grundlehren der mathematischen Wissenschaften 153, Springer-Verlag (1969).Google Scholar
  12. [Fu1]K. Fukaya, Collapsing Riemannian manifolds to ones of lower dimensions, J. Differential Geom. 25 (1987), 139–156.Google Scholar
  13. [Fu2]K. Fukaya, Collapsing Riemannian manifolds and eigenvalues of the Laplace operator, Invent. Math. 87 (1987), 517–547.Google Scholar
  14. [Fu3]K. Fukaya, A boundary of the set of Riemannian manifolds with bounded curvature and diameter, J. Diff. Geom. 28:1 (1988), 1–21.Google Scholar
  15. [Fu4]K. Fukaya, Collapsing Riemannian manifolds to ones of lower dimension, II, Math. Soc. Japan 41:2 (1989), 333–356.Google Scholar
  16. [G]S. Gallot, Inégalités isopérimétriques et analytiques sur les variétés riemanniennes, S.M.F. Astérisque 163/164 (1988), 31–91.Google Scholar
  17. [Gr]M. Gromov, Filling Riemannian manifolds, J. Differential Geom. 18 (1983), 1–147.Google Scholar
  18. [GrLP]M. Gromov, J. Lafontaine, P. Pansu, Structures métriques pour les variétés Riemanniennes, Cédic-Fernand Nathan (1981).Google Scholar
  19. [HSU]H. Hess, R. Schrader, D.A. Uhlenbrock, Kato's inequality and the spectral distribution of Laplacians on compact Riemannian manifolds, J. Diff. Geom. 15 (1980), 27–38.Google Scholar
  20. [KKu]A. Kasue, H. Kumura, Spectral convergence of Riemannian manifolds, preprint (1992).Google Scholar
  21. [L]H.B. Lawson, Lectures on minimal submanifolds, Lecture series 9, Publish or Perish Inc., Berkeley (1980).Google Scholar
  22. [Mu]H. Muto, On the spectral distance, preprint (1988).Google Scholar

Copyright information

© Birkhäuser Verlag 1994

Authors and Affiliations

  • P. Bérard
    • 1
  • G. Besson
    • 1
  • S. Gallot
    • 2
  1. 1.Institut FourierUniversité de Grenoble 1St Martin d'HèresFrance
  2. 2.École Normale Supérieure de LyonLyonFrance

Personalised recommendations